Regularization effect on autoencoder

import time
from typing import Tuple, Dict, Union, List

import tensorflow as tf
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import numpy as np
from sklearn.manifold import TSNE
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_samples, silhouette_score

NUMBER_PARAM = 10
REGULARIZER_TYPE = 'kernel_regularizer'
LS_KERNEL_REGULARIZER = np.linspace(0, 1e-4, NUMBER_PARAM)
LS_ACTIVITY_REGULARIZER = np.zeros(NUMBER_PARAM)

1. Data prep

(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()

x_train_norm = x_train/np.max(x_train)
x_test_norm = x_test/np.max(x_test)

2. Model architecture

The following model is to minimize loss function, $L_T$, in terms of loss from regular autoencoder, $L$, plus regularization term, $R$.

Encoding layer:
$h = \alpha_e(W_1 \times\ x + b_1)$, where $\alpha_e(.)$ is activation function ReLU, and the number of hidden units in $h$ is 196. Hence $h$ is a 196x1 vector in 196-dim latent space, $W_1$ is 196x748 weight matrix and $b_1$ is the bias term in the form of 196x1 vector.
Decoding layer:
$x' = \alpha_d(W_2 \times\ h + b_2)$, where $\alpha_d(.)$ is activation function sigmoid, $x'$ is the output of the autoencoder which is optimized to reconstruct back to input $x$.
Loss function:
$L_T = L + R = ||x - x'||^2 + \lambda_k\sum |W_1|^2 + \lambda_k\sum |W_2|^2$, where $i$ is the number of hidden units, $\lambda_k\ $is kernel regularizer, l2 regularization is used.

class Autoencoder(tf.keras.Model):
    def __init__(
        self, 
        input_shape: Tuple[int,int], 
        encoding_dim: int,
        activity_regularizer: float,
        kernel_regularizer: float,
        **kwargs
    ) -> None:
        
        super().__init__(**kwargs)
        self.encoding_dim = encoding_dim
        self.encoder = tf.keras.Sequential([
            tf.keras.layers.Flatten(),
            tf.keras.layers.Dense(
                encoding_dim, 
                activation='relu', 
                activity_regularizer=tf.keras.regularizers.L2(activity_regularizer),
                kernel_regularizer=tf.keras.regularizers.L2(kernel_regularizer)
            )
        ])
        self.decoder = tf.keras.Sequential([
          tf.keras.layers.Dense(
              input_shape[0]*input_shape[1], 
              activation='sigmoid',
              kernel_regularizer=tf.keras.regularizers.L2(kernel_regularizer)
          ),
          tf.keras.layers.Reshape(input_shape)
        ])
        
    def call(self, inputs):
        encoded = self.encoder(inputs)
        decoded = self.decoder(encoded)
        return decoded

3. Training

def training(
    train_set: np.ndarray,
    test_set: np.ndarray,
    activity_regularizer: float,
    kernel_regularizer: float,
    encoding_dim: int,
) -> Dict[str, Union[Autoencoder, float, int, np.ndarray]]:
    
    # build
    autoencoder = Autoencoder(
        input_shape=(28, 28),
        encoding_dim=encoding_dim,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
    )
    autoencoder.compile(optimizer='adam', loss='mse')
    callback = tf.keras.callbacks.EarlyStopping(monitor='val_loss', patience=3, min_delta=0.0001)

    # fit
    print(f'Training start:')
    print(f'activity_regularizer = {activity_regularizer}')
    print(f'kernel_regularizer = {kernel_regularizer}')
    start = time.time()
    history = autoencoder.fit(
        train_set, train_set,
        callbacks=[callback],
        epochs=100,
        batch_size=256,
        shuffle=True,
        verbose=0,
        validation_data=(test_set, test_set)
    )
    end = time.time()

    # eval
    training_loss = history.history.get('loss')[-1]
    testing_loss = history.history.get('val_loss')[-1]
    epoch = max(history.epoch)
    print('Training results:')
    print(f'training_loss = {training_loss}')
    print(f'testing_loss = {testing_loss}')
    print(f'epoch = {epoch}')
    print(f'time passed = {int(round(end-start))}s')
    print('-'*100)
    
    # return
    w1, _, _, _ = autoencoder.get_weights()
    w1_reshape = w1.T.reshape((encoding_dim,28,28))
    result = {
        'hyperparam': {
            'activity_regularizer': activity_regularizer,
            'kernel_regularizer': kernel_regularizer,
        },
        'results': {
            'training_loss': training_loss, 
            'testing_loss': testing_loss, 
            'epoch': epoch,
            'model': autoencoder,
            'w1_reshape': w1_reshape,
        }
    }
    
    return result

training_results = list()
for kernel_regularizer, activity_regularizer in zip(LS_KERNEL_REGULARIZER,LS_ACTIVITY_REGULARIZER):
    result = training(
        train_set=x_train_norm, 
        test_set=x_test_norm,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
        encoding_dim=196
    )
    training_results.append(result)

Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0
Training results:
training_loss = 0.0014675884740427136
testing_loss = 0.001450086128897965
epoch = 24
time passed = 35s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 1.1111111111111112e-05
Training results:
training_loss = 0.014790416695177555
testing_loss = 0.014556732028722763
epoch = 17
time passed = 26s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 2.2222222222222223e-05
Training results:
training_loss = 0.021235274150967598
testing_loss = 0.020908936858177185
epoch = 17
time passed = 24s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 3.3333333333333335e-05
Training results:
training_loss = 0.02563323639333248
testing_loss = 0.025231678038835526
epoch = 23
time passed = 31s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 4.4444444444444447e-05
Training results:
training_loss = 0.029257353395223618
testing_loss = 0.028820205479860306
epoch = 23
time passed = 35s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 5.555555555555556e-05
Training results:
training_loss = 0.03215980529785156
testing_loss = 0.03170749545097351
epoch = 26
time passed = 38s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 6.666666666666667e-05
Training results:
training_loss = 0.03472293168306351
testing_loss = 0.03426063805818558
epoch = 26
time passed = 36s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 7.777777777777778e-05
Training results:
training_loss = 0.037063032388687134
testing_loss = 0.03661523759365082
epoch = 23
time passed = 32s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 8.888888888888889e-05
Training results:
training_loss = 0.03865893930196762
testing_loss = 0.038147132843732834
epoch = 33
time passed = 45s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0001
Training results:
training_loss = 0.04047049209475517
testing_loss = 0.03999369964003563
epoch = 30
time passed = 40s
----------------------------------------------------------------------------------------------------

4. Loss plot

MSE loss ($L_T$) against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def loss_plot(results) -> None:
    param = list()
    training_loss = list()
    testing_loss = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        training_loss.append(result['results']['training_loss'])
        testing_loss.append(result['results']['testing_loss'])

    plt.figure(figsize=(12, 8))
    plt.plot(param, testing_loss, 'bs', label='testing_loss')
    plt.plot(param, training_loss, 'r^', label='training_loss')
    plt.ylabel('loss')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training loss vs testing loss')
    plt.legend(loc='upper left')
    plt.show()
    plt.close()

loss_plot(results=training_results)

5. Sparsity plot

Sparsity of $h$ against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def _sparsity_map(x: int) -> None:
    if x != 0:
        return 1
    else:
        return 0
    
_sparsity_map_vec = np.vectorize(_sparsity_map)

def plot_sparsity(
    results, 
    train_set, 
    test_set
) -> None:
    param = list()
    ls_training_sparsity = list()
    ls_testing_sparsity = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        model = result['results']['model']
        train_encoded_imgs = model.encoder(train_set).numpy()
        test_encoded_imgs = model.encoder(test_set).numpy()
        train_sparsity = np.sum(_sparsity_map_vec(train_encoded_imgs))/train_encoded_imgs.size
        testing_sparsity = np.sum(_sparsity_map_vec(test_encoded_imgs))/test_encoded_imgs.size
        ls_training_sparsity.append(train_sparsity)
        ls_testing_sparsity.append(testing_sparsity)
        
    plt.figure(figsize=(12, 8))
    plt.plot(param, ls_testing_sparsity, 'bs', label='testing_sparsity')
    plt.plot(param, ls_training_sparsity, 'r^', label='training_sparsity')
    plt.ylabel('sparsity')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training sparsity vs testing sparsity')
    plt.legend(loc='upper right')
    plt.show()
    plt.close()

plot_sparsity(
    results=training_results,
    train_set=x_train_norm,
    test_set=x_test_norm,
)

6. Weight matrix of the encoder

Weight matrix of the encoder, $W_1$, is shown on a grey-scale heatmap. Each of the subplot showing a row frm $W_1$ reshaped to 28x28

def plot_w1(
    w1: np.ndarray
) -> None:
    
    w1_dim = int(np.sqrt(len(w1)))
    fig, ax = plt.subplots(
        nrows=w1_dim, 
        ncols=w1_dim, 
        figsize=(w1_dim,w1_dim)
    )
    plt.gray()

    i = 0
    for row in ax:
        for col in row:
            col.imshow(w1[i])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            i = i + 1
            
    plt.show()
    plt.close()

for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    plot_w1(
        w1=result['results']['w1_reshape']
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

7. Original image vs latent space vs reconstructed image

def plot_images(
    img_gp: List[np.ndarray],
    num_img: int,
) -> None:
    
    _, ax = plt.subplots(nrows=len(img_gp), ncols=num_img, figsize=(num_img,len(img_gp)))
    plt.gray()
    for i, row in enumerate(ax):
        tmp = img_gp[i]
        for j, col in enumerate(row):
            col.imshow(tmp[j])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            
    plt.show()
    plt.close()

num_img = 20
img = x_test_norm[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    decoded_img = model.decoder(encoded_img).numpy()
    latent_dim = int(np.sqrt(len(encoded_img.T)))
    encoded_img_reshape = encoded_img.reshape((num_img,latent_dim,latent_dim))
    norm_encoded_img_reshape = norm_encoded_img.reshape((num_img,latent_dim,latent_dim))
    
    img_gp = [img, encoded_img_reshape, norm_encoded_img_reshape, decoded_img]
    plot_images(
        img_gp=img_gp,
        num_img=num_img,
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

8. Latent space similarity plot

def similarity_plot(
    img: np.ndarray, 
    ls_label: List[int]
) -> None:
    
    similarity_matrix = img @ img.T
    ls_similarity = [list() for i in range(10)]
    ls_disimilarity = [list() for i in range(10)]
    for i, u in enumerate(similarity_matrix):
        for j, v in enumerate(u):
            if i == j:
                continue
            if ls_label[i] == ls_label[j]:
                label = ls_label[i]
                ls_similarity[label].append(v)
            else:
                label = ls_label[i]
                ls_disimilarity[label].append(v)
    
    plot_data = [
        {
            'data': ls_similarity,
            'title': 'Cosine similarity of the latent vector with the same label'
        },
        {
            'data': ls_disimilarity,
            'title': 'Cosine similarity of the latent vector with the different label',
        }
    ]
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(24,8))
    for idx, col in enumerate(ax):
        col.boxplot(plot_data[idx]['data'])
        col.set_title(plot_data[idx]['title'])
        col.set_ylim([0,1])
        col.set_xticks(range(1,11))
        col.set_xticklabels(range(10))
        col.set_xlabel('label')
        col.set_ylabel('cosine similarity')
        
    plt.show()
    plt.close()

num_img = 1000
img = x_test_norm[:num_img]
img_label = y_test[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    similarity_plot(img=norm_encoded_img, ls_label=img_label)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

9. Latent space in T-SNE space

def tsne_plot(
    space_gp: List[np.ndarray],
    label: List[int]
) -> None:
    
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(20,10))
    for idx, col in enumerate(ax):
        space = space_gp[idx]['space']
        scatter = col.scatter(space[:,0], space[:,1], c=label, cmap='Spectral')
        col.set_title(space_gp[idx]['title'])
        col.set_xlabel('tsne 1')
        col.set_ylabel('tsne 2')
        col.legend(*scatter.legend_elements())
    plt.show()
    plt.close()

for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    print('fitting tsne for encoded_img')
    tsne_space = TSNE(n_components=2, n_jobs=-1).fit_transform(encoded_img)
    print('fitting tsne for norm_encoded_img')
    tsne_space_norm = TSNE(n_components=2, n_jobs=-1).fit_transform(norm_encoded_img)
    space_gp = [
        {
            'space': tsne_space,
            'title': 'Latest space',
        },
        {
            'space': tsne_space_norm,
            'title': 'Normalized latest space',
        },
    ]
    tsne_plot(space_gp=space_gp, label=y_test)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

10. K-Mean plot

def kmean_plot(
    img: np.ndarray,
    max_n_cluster: int = 10,
) -> None:
    
    _, ax = plt.subplots(nrows=2, ncols=5, figsize=(25,12))
    ax = ax.flatten()
    range_clusters = range(2,max_n_cluster+1)
    for n_clusters in range_clusters:
        kmeans = KMeans(n_clusters=n_clusters)
        kmeans.fit(img)
        labels = kmeans.labels_
        avg_silhouette_score = silhouette_score(img, labels, metric='euclidean')
        sample_silhouette_values = silhouette_samples(img, labels)
        y_lower = 10
        for i in range(n_clusters):
            ith_cluster_silhouette_values = sample_silhouette_values[labels == i]
            ith_cluster_silhouette_values.sort()
            size_cluster_i = ith_cluster_silhouette_values.shape[0]
            y_upper = y_lower + size_cluster_i
            color = cm.nipy_spectral(float(i) / n_clusters)
            ax[n_clusters-1].fill_betweenx(
                np.arange(y_lower, y_upper),
                0, ith_cluster_silhouette_values,
                facecolor=color, 
                edgecolor=color, 
                alpha=0.7
            )
            ax[n_clusters-1].set_xlim([-0.1, 1])
            ax[n_clusters-1].set_ylim([0, len(img) + (n_clusters + 1) * 10])
            ax[n_clusters-1].text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
            ax[n_clusters-1].set_xlabel('silhouette score')
            ax[n_clusters-1].set_ylabel('cluster label')
            ax[n_clusters-1].set_title(f'{n_clusters}-clusters')
            ax[n_clusters-1].axvline(x=avg_silhouette_score, color="red", linestyle="--")
            ax[n_clusters-1].set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
            ax[n_clusters-1].set_yticks([])
            y_lower = y_upper + 10

    plt.show()
    plt.close()

ls_inertia = list()
for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    space_gp = [
        {
            'space': encoded_img,
            'title': 'Latest space',
        },
        {
            'space': norm_encoded_img,
            'title': 'Normalized latest space',
        },
    ]
    for space in space_gp:
        print(f'img = {space["title"]}')
        kmean_plot(img=space['space'])

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
img = Latest space

img = Normalized latest space